Measurement method for isoseismal major and minor axes based on ellipse fitting algorithm
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摘要:
提出了一种改进的拟合椭圆算法用于测量等震线长短轴，即将等震线测量的模式和直接测量算法的概念以约束的形式加入拟合椭圆算法，并使用最小二乘法求解椭圆系数，在与传统等震线测量方法保持一致的基础上，减少了直接测量方法的主观不确定性。在此基础上讨论了拟合椭圆算法的适用性和鲁棒性，将等震线长短轴的测量结果与前人的结果进行了对比。结果表明：拟合椭圆算法的适用性较好，可用于等震线的拟合；拟合椭圆算法的鲁棒性较好，计算结果较为稳定；对于未闭合且形状复杂的等震线，计算结果相对离散，要单独验证结果是否合理；利用拟合椭圆算法得到的等震线长短轴结果与前人的结果较为一致，可用于建立烈度衰减关系。
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关键词:
- 等震线长短轴测量方法 /
- 拟合椭圆算法 /
- 最小二乘拟合 /
- 算法测试
Abstract:As the fundamental data for establishing the seismic intensity attenuation relationship, the reliability of isoseismal data directly affects the accuracy and reliability of the intensity attenuation relationship. The acquisition of traditional isoseismal parameters is based on direct measurement, which inevitably introduces a degree of subjectivity. A more objective approach to measuring the parameters of isoseismals has been proposed. However, it is inevitable that conceptual differences will exist due to algorithmic differences and the lack of links between the novel algorithm and commonly used direct measurement methods. In view of this, this paper introduces a novel approach to measuring the major and minor axes of isoseismals. The mode of isoseismal measurement and the concepts from the direct measurement method are incorporated into the ellipse fitting algorithm as mathematical constraints, thereby providing a systematic method for measuring the radii of isoseismal major and minor axes. In order to reduce the subjective influence of isoseismal major and minor-axis measurements, the radii of isoseismal major and minor axes are measured using the best-fitting ellipse of the isoseismal. This approach is based on the consistency and inheritance of the methods and concepts of the direct measurements. The resulting data provide the basic data for the establishment of intensity attenuation relationship.
The measurement modes of isoseismals are classified into two types according to whether the major-axis strike is fixed or not. In order to accommodate this distinction, the strike of major axis is also incorporated into the algorithm as a constraint. Additionally, the area of isoseismal is incorporated into the algorithm as a constraint. This leads to the proposal of four ellipse fitting algorithms in total: ① An unconstrained fitting algorithm, which implies that no additional constraints are imposed during the model fitting process. ② A fitting algorithm with major-axis strike constrained, which involves the addition of a priori major-axis strike information to the model fitting process. ③ A fitting algorithm with area and major-axis strike constrained, which incorporates both a priori major-axis strike and area into the model fitting process. ④ A fitting algorithm with area constrained, which entails the addition of a priori model area to the model fitting process.
It should be noted that the results of the algorithm will degrade from an ellipse to a circle when the isoseismal shape is circular （i.e. a special ellipse）. Therefore, it is necessary to construct circular isoseismal and simulate different levels of sampling noises so as to test the applicability of the algorithm. The results of the algorithm test demonstrate that all four algorithms yield satisfactory outcomes when the noise is minimal. However, the algorithm with area constraints exhibits a significantly superior performance compared with the algorithm without area constraints when the noise is elevated. Nevertheless, the applicability of the algorithms for degradation to a circle is deemed acceptable, given that extreme noise cases are unlikely to occur in practical scenarios.
The isoseismal data used by the ellipse fitting algorithm is subject to sampling bias, which may affect the output of algorithm. Therefore, it is necessary to analyze the impact of data sampling differences on the results of algorithm. The robustness of the algorithm was evaluated by randomly sampling points of isoseismal to simulate different sampling scenarios. The results indicate that the parameters show normally distributed, with the majority falling within one standard deviation of the mean value. Furthermore, the algorithm is robust. For the unclosed isoseismal with complex shapes, the calculation results are relatively discrete, and thus require separate verification. Furthermore, the measurements obtained by the ellipse fitting algorithm are consistent with previous results and can be used to establish the intensity attenuation relationship.
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图 1 椭圆长轴走向示意图
（a） 走向范围为0°—90°；（b） 走向范围为−90°—0°
Figure 1. Diagram of the major-axis strike of the ellipse
（a） Strike ranges 0°−90°；（b） Strike ranges −90°−0°
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图 2 长轴转向（左）和不转向（右）模式的地震等震线图拟合结果
（a，b） 1976年7月28日唐山MS7.8地震；（c，d） 2008年5月12日汶川MS8.0地震
Figure 2. Fitting results of the isoseismals of the earthquakes with major-axis steering （a） and non-steering （b） modes
（a，b） Tangshan MS7.8 earthquake on July 28，1976； （c，d） Wenchuan MS8.0 earthquake on May 12，2008
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图 3 代表性的等震线
（a） 简单闭合等震线：2013年芦山MS7.0地震Ⅶ度等震线；（b） 复杂闭合等震线：1945年滦县MS6.3地震Ⅶ度等震线；（c） 简单未闭合等震线：1986年阳江MS4.8地震Ⅳ度等震线；（d） 复杂未闭合等震线：1976年唐山MS7.8地震Ⅵ度等震线
Figure 3. Representative isoseismals
（a） Closed isoseismal with simple shape：Isoseismal Ⅶ of 2013 Lushan MS7.0 earthquake；（b） Closed isoseismal with complex shape：Isoseismal Ⅶ of 1945 Luanxian MS6.3 earthquake；（c） Unclosed isoseismal with simple shape：Isoseismal Ⅳ of 1986 Yangjiang MS4.8 earthquake；（d） Unclosed isoseismal with complex shape：Isoseismal Ⅵ of 1976 Tangshan MS7.8 earthquake
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图 4 四类等震线在不同无约束算法下的拟合结果的对比
（a） 简单闭合等震线：2013年芦山MS7.0地震Ⅶ度等震线；（b） 复杂闭合等震线：1945年滦县MS6.3地震Ⅶ度等震线；（c） 简单未闭合等震线：1986年阳江MS4.8地震Ⅳ度等震线；（d） 复杂未闭合等震线：1976年唐山MS7.8地震Ⅵ度等震线
Figure 4. Comparison of the fitting results of four isoseismals by the different unconstrained algorithms
（a） Closed isoseismal with simple shape：Isoseismal Ⅶ of 2013 Lushan MS7.0 earthquake；（b） Closed isoseismal with complex shape：Isoseismal Ⅶ of 1945 Luanxian MS6.3 earthquake；（c） Unclosed isoseismal with simple shape：Isoseismal Ⅳ of 1986 Yangjiang MS4.8 earthquake；（d） Unclosed isoseismal with complex shape：Isoseismal Ⅵ of 1976 Tangshan MS7.8 earthquake
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图 5 四类等震线在无约束拟合算法和约束长轴走向拟合算法下拟合结果的对比
（a） 简单闭合等震线：2013年芦山MS7.0地震Ⅶ度等震线；（b） 复杂闭合等震线：1945年滦县MS6.3地震Ⅶ度等震线；（c） 简单未闭合等震线：1986年阳江MS4.8地震Ⅳ度等震线；（d） 复杂未闭合等震线：1976年唐山MS7.8地震Ⅵ度等震线
Figure 5. Comparison of the fitting results of the isoseismals by the unconstrained algorithm with those by the algorithm with major-axis strike constrained
（a） Closed isoseismal with simple shape：Isoseismal Ⅶ of 2013 Lushan MS7.0 earthquake；（b） Closed isoseismal with complex shape：Isoseismal Ⅶ of 1945 Luanxian MS6.3 earthquake；（c） Unclosed isoseismal with simple shape：Isoseismal Ⅳ of 1986 Yangjiang MS4.8 earthquake；（d） Unclosed isoseismal with complex shape：Isoseismal Ⅵ of 1976 Tangshan MS7.8 earthquake
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图 6 遍历算法1和遍历算法2计算过程及结果对比
（a） 2013年4月20日芦山MS7.0地震Ⅶ度等震线；（b） 1945年9月23日滦县MS6.3地震Ⅶ度等震线
Figure 6. Comparison of calculation process and result of two traversal algorithms
（a） Isoseismal Ⅶ of Lushan MS7.0 earthquake on April 20，2013；（b） Isoseismal Ⅶ of Luanxian MS6.3 earthquake on September 23，1945
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图 7 约束长轴走向的拟合算法和约束面积与长轴走向的拟合算法所得拟合结果的对比
（a） 2013年4月20日芦山MS7.0地震Ⅶ度等震线；（b） 1945年9月23日滦县MS6.3地震Ⅶ度等震线
Figure 7. Comparison of fitting results obtained by the algorithm with major-axis strike constrained with those by the algorithm with area and major-axis strike constrained
（a） Isoseismal Ⅶ of Lushan MS7.0 earthquake on April 20，2013；（b） Isoseismal Ⅶ of Luanxian MS6.3 earthquake on September 23，1945
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图 8 无约束拟合算法和约束面积的拟合算法所得拟合结果的对比
（a） 2013年4月20日芦山MS7.0地震Ⅶ度等震线；（b） 1945年9月23日滦县MS6.3地震Ⅶ度等震线
Figure 8. Comparison of fitting results obtained by the unconstrained algorithm with those by the algorithm with area constrained
（a） Isoseismal Ⅶ of Lushan MS7.0 earthquake on April 20，2013；（b） Isoseismal Ⅶ of Luanxian MS6.3 earthquake on September 23，1945
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图 9 误差水平σ＝20时四种算法的适用性检验
（a） 无约束拟合算法；（b） 约束长轴走向的拟合算法；（c） 约束面积与长轴走向的拟合算法；（d） 约束面积的拟合算法
Figure 9. Applicability test of four algorithms when error σ is taken as 20
（a） Unconstrained fitting algorithm；（b） Fitting algorithm with major-axis strike constrained；（c） Fitting algorithm with area and major-axis strike constrained；（d） Fitting algorithm with area constrained
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图 10 误差水平σ＝100时四种算法的适用性检验
（a） 无约束拟合算法；（b） 约束长轴走向的拟合算法；（c） 约束面积与长轴走向的拟合算法；（d） 约束面积的拟合算法
Figure 10. Applicability test of four algorithms when error σ is taken as 100
（a） Unconstrained fitting algorithm；（b） Fitting algorithm with major-axis strike constrained；（c） Fitting algorithm with area and major-axis strike constrained；（d） Fitting algorithm with area constrained
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12 算法鲁棒性检验（参数结果）
灰色三角形为每次抽样计算的结果，红色虚线为抽样计算参数的一倍标准差，红色实线为抽样计算的平均结果，黑色虚线为全部数据的计算结果，直方图给出参数计算结果的统计情况（a） 无约束拟合算法；（b） 约束长轴走向的拟合算法
12. Robustness test of the four algorithms （parameter results）
The grey triangle is the result of each sampling calculation，the dashed red line is one standard deviation of the sampling calculation，the solid red line is the average result of sampling calculation，the black dashed line is the calculation result of the overall data，and the histogram gives the statistics of the parameter calculations（a） Unconstrained fitting algorithm；（b） Fitting algorithm with major-axis strike constrained
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12 算法鲁棒性检验（参数结果）
灰色三角形为每次抽样计算的结果，红色虚线为抽样计算参数的一倍标准差，红色实线为抽样计算的平均结果，黑色虚线为全部数据的计算结果，直方图给出参数计算结果的统计情况（c） 约束面积与长轴走向的拟合算法；（d） 约束面积的拟合算法
12. Robustness test of the four algorithms （parameter results）
The grey triangle is the result of each sampling calculation，the dashed red line is one standard deviation of the sampling calculation，the solid red line is the average result of sampling calculation，the black dashed line is the calculation result of the overall data，and the histogram gives the statistics of the parameter calculations（c） Fitting algorithm with area and major-axis strike constrained；（d） Fitting algorithm with area constrained
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图 13 算法鲁棒性检验的结果示意图
（a） 无约束拟合算法；（b） 约束长轴走向的拟合算法；（c） 约束面积与长轴走向的拟合算法；（d） 约束面积的拟合算法
Figure 13. Robustness test of the four algorithms （diagram of the results）
（a） Unconstrained fitting algorithm；（b） Fitting algorithm with major-axis strike constrained；（c） Fitting algorithm with area and major-axis strike constrained；（d） Fitting algorithm with area constrained
表 1 1976年7月28日唐山MS7.8地震等震线测量结果对比
Table 1 Comparison of isoseismal measurement results of MS7.8 Tangshan earthquake on July 28，1976
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表 2 2008年5月12日汶川MS8.0地震等震线测量结果对比
Table 2 Comparison of isoseismal measured by MS8.0 Wenchuan earthquake on May 12，2008
等震线
烈度本文 肖亮和俞言祥（2 011） 长轴/km 短轴/km 长轴/km 短轴/km XI 33.61 10.15 33 10 XI 44.51 7.34 44 7.5 X 121.36 14.15 112.1 14 IX 161.47 25.24 159 22.5 VIII 212.09 60.41 206.5 57.5 VII 280.26 140.76 283 133.5 VI 471.23 295.57 468 298
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